atom-atom interactions in ultracold quantum gasescastin/cohen1.pdf · 1 collège de france...

1Collège de France

Atom-Atom Interactions in Ultracold Quantum Gases

Claude Cohen-Tannoudji

Lectures on Quantum GasesInstitut Henri Poincaré, Paris, 25 April 2007

2

Lecture 1 (25 April 2007)

Quantum description of elastic collisions between ultracold atoms

The basic ingredients for a mean-field description of gaseous Bose Einstein condensates

Lecture 2 (27 April 2007)

Quantum theory of Feshbach resonances

How to manipulate atom-atom interactions in a ultracold quantum gas

3

A few general references

1 – L.Landau and E.Lifshitz, Quantum Mechanics, Pergamon, Oxford (1977)2 – A.Messiah, Quantum Mechanics, North Holland, Amsterdam (1961)3 – C.Cohen-Tannoudji, B.Diu and F.Laloë, Quantum Mechanics, Wiley,

New York (1977)4 – C.Joachin, Quantum collision theory, North Holland, Amsterdam (1983)5 – J.Dalibard, in Bose Einstein Condensation in Atomic Gases, edited by

M.Inguscio, S.Stringari and C.Wieman, International School of PhysicsEnrico Fermi, IOS Press, Amsterdam, (1999)

6 – Y. Castin, in ’Coherent atomic matter waves’, Lecture Notes of Les Houches Summer School, edited by R. Kaiser, C. Westbrook, and F. David, EDP Sciences and Springer-Verlag (2001)

7 – C.Cohen-Tannoudji, Cours au Collège de France, Année 1998-1999http://www.phys.ens.fr/cours/college-de-france/

9 – T.Köhler, K.Goral, P.Julienne, Rev.Mod.Phys. 78, 1311-1361 (2006)

8 – C.Cohen-Tannoudji, Compléments de mécanique quantique, Cours de3ème cycle, Notes de cours rédigées par S.Harochehttp://www.phys.ens.fr/cours/notes-de-cours/cct-dea/index.html/

http://www.phys.ens.fr/cours/college-de-france/

http://www.phys.ens.fr/cours/notes-de-cours/cct-dea/index.html

4

Outline of lecture 11 - Introduction2 - Scattering by a potential. A brief reminder

• Integral equation for the wave function• Asymptotic behavior. Scattering amplitude• Born approximation

3 - Central potential. Partial wave expansion• Case of a free particle• Effect of the potential. Phase shifts• S-Matrix in the angular momentum representation

4 - Low energy limit• Scattering length a• Long range effective interactions and sign of a

5 - Model used for the potential. Pseudo-potential• Motivation• Determination of the pseudo-potential• Scattering and bound states of the pseudo-potential• Pseudo-potential and Born approximation

5

Interactions between ultracold atomsAt low densities, 2-body interactions are predominant and can be described in terms of collisions. We will focus here on elastic collisions (although inelastic collisions and 3-body collisions are also important because they limit the achievable spatial densities of atoms).Collisions are essential for reaching thermal equilibriumAt very low temperatures, mean-field descriptions of degenerate quantum gases depend only on a very small number of collisionalparameters. For example, the shape and the dynamics of Bose Einstein condensates depend only on the scattering lengthPossibility to control atom-atom interactions with Feshbach resonances. This explains the increasing importance of ultracoldatomic gases as simple models for a better understanding of quantum many body systems Purpose of these lectures: Present a brief review of the concepts of atomic and molecular physics which are needed for a quantitativedescription of interactions in ultracold atomic gases.

6

Notation

1 2( )V r r−Two atoms, with mass m, interacting with a 2-body interaction potential In lecture 1, we ignore the spins degrees of freedom. They will be taken into account in lecture 2.Hamiltonian 2 2

1 21 22 2

( )p p

H V r rm m

= + + −

Change of variables

( )1 2 1 2

1 2

22

/G GR r r P p pM m m m

= + = += + =

Center of mass variablesTotal mass

( )( )

1 2 1 2

1 2 1 2

22

Relative variables Reduced massμ

= − = −= + =

/

/ /

r r r p p pm m m m m

( ) ( )2 22 2μ= + +/ /

CM rel

G

H H

H P M p V r

Hamiltonian of a free particle with mass M

Hamiltonian of a “fictitious” particle with mass μ, moving in V(r)

(1.1)

(1.2)

7

Finite range potential

One can extend the results obtained in this simple case to potentials decreasing fast enough with r at large distances.For example, for the Van der Waals interactions between atoms decreasing as C6 / r6 for large r, one can define an effective range

( ) 0V r r bb

= >Simple case where for is called the range of the potential

1 4

62

2VdW

μ⎛ ⎞= ⎜ ⎟

⎝ ⎠

/C

b

See for example Ref. 5

(1.3)

8





5 - Model used for the potential. Pseudo-potential• Motivation• Determination of the pseudo-potential• Scattering and bound states of the pseudo-potential

9

Scattering by a potential. A brief reminderShrödinger equation for the relative particle (with E>0)

( )2 2 2

( ) ( )2 2

kV r r E r Eψ ψμ μ

⎡ ⎤− Δ + = =⎢ ⎥

⎣ ⎦

( ) ( ) ( )22

2μψ ψ⎡ ⎤Δ + =⎣ ⎦k r V r r

Green function of Δ + k2

( ) ( )2 δ⎡ ⎤Δ + =⎣ ⎦k G r rThe boundary conditions for G will be chosen later on

Integral equation for the solution of Shrödinger equation( )

( )0

20 0

Solution of the equation without the right memberϕ

ϕ⎡ ⎤Δ + =⎣ ⎦

:r

k r

( ) ( ) ( ) ( ) ( )30 dψ ϕ ψ′ ′ ′ ′= + −∫r r r G r r V r r

(1.7)

(1.4)

(1.5)

(1.6)

(1.8)

10

Choice of boundary conditionsWe choose for ϕ0 a plane wave with wave vector k

( )0 e e κϕ κ= = =. . /i k r i k rr k kand we choose, for the Green function G, boundary conditions corresponding to an outgoing spherical wave (see Ref.2, Chap.XIX)

14

eπ

′−

+ ′− = −′−

( )i k r r

G r rr r

We thus get the following solution for Schrödinger equation31

4ee dψ ψ

π

′−+ +′ ′ ′= −

′−∫.() ()()i k r r

i k rk kr r V r r

r rIf V has a finite range b, the integral over r’ is restricted to a finite range and we can write:

32

24

ψ κ

μκ ψπ

+

′ +

′ ′− − =

−

′ ′ ′= − ∫

.

- .

( ) ( , ,

, .

)

( , , ) ( ) ( )

/If withee

d e

i kri k r

k

i k n rk

r f k nr

f k n

r b r r r r n

r V r

n r

r

r

(1.9)

(1.10)

(1.11)

(1.12)

11

Scattering state with an outgoing spherical wave

Asymptotic behavior for large r

Scattering amplitude f

Differential cross section

( )The state is a solution of the Schrodinger equation behavingfor large r as the sum of an incoming plane wave and ofan outgoing spherical wave

ψ

κ

+()

exp.

(,,)exp( )/

k ri k r

f k n i k r r

32

24

We have pu

e

t

dμκ ψπ

′ ′ +′ ′ ′=

′

−

= =∫ -.(,,) () )

/

(i k rkf k n r

k k n k

V r

r

r

r(,,)

/

f k nk k n k r r k

k k

κ

θ ϕ′ = =

′

is the amplitude of the outgoing spherical wave in thedirection of . It depends only on and on the polar angles and of with respect to

Comparing the fluxes along k and k’, one gets :2

d dσ κΩ =/ (,,)f k n (1.14)

(1.13)

12

Born approximation

32

24

d eμκ ψπ

′ ′ +′ ′ ′= − ∫ -.(,,) ()()i k rkf k n r V r r

()

exp(.)

k r

i k r

ψ + ′

In the scattering amplitude, the potential V appears explicitly

To lowest order in V, one can thus replace by the zeroth

order solution of the Schrodinger equation

32

24

d eμκπ

′ ′−′ ′= − ∫ ().(,,) ()i k k rf k n r V r

This is the Born approximation

In this approximation, the scattering amplitude is proportional to the spatial Fourier transform of the potential

(1.15)

(1.16)

13

Low energy limitThe presence of V(r’) in the scattering amplitude

32

24

-.(,,) ()()i k rkf k n r V r rμκ ψ

π′ ′ +′ ′ ′= − ∫ d e

restricts the integral over r’ to a finite range r’< b

32

124

If , one can replace e by 1. The scattering amplitude

d

then no longer depends on the direction of the scattering vector It is spherically symmetric

μκ ψπ

′ ′−

+′ ′ ′= −

′∫

.

( , , ) ( ) ( )

.

i k r

k

kb

f k n r V r r

k even if is not.( )V r

0

1

When ee

κ

ψ +

→ → −

− → −.

,(,,)

()i k r

i k rk

k f k n aar a

r r

a is a constant, called “scattering length”, which will be discussed in more details later on

(1.17)

(1.18)

(1.19)

14

Another interpretation of the outgoing scattering state

Another expression for this state (see refs. 4 and 8)

Ingoing scattering state

2

0

1 2Limε

ψ ϕ ψ με+

+ +

→= + =

− +/k k kV T p

E T i

0For non zero but very small, appears as the state obtainedat by starting from the free state at and by switching on slowly on a time interval on the order of

k

kt tV

ε ψϕ

ε

+

= = −∞/

3

0

14

1

ee d

Limε

ψ ψπ

ψ ϕ ψε+

′− −− −

− −

→

′ ′ ′= −′−

= +− −

∫.() () ()

i k r ri k r

k k

k k k

r r V r rr r

VE T i

0If one starts from such a state at and if one switchesoff slowly on a time interval on the order of , one getsthe free state at

εϕ

=

= +∞/

k

tV

t

(1.20)

(1.21)

15

S - MatrixDefinition

1

2

2 1Lim

evolution operator in interaction representation

ϕ ϕ ϕ ϕ→−∞→+∞

= =,

(,)

:

j i j iji k k k ktt

S S U t t

U

One can show that ψ ψ− +=j iji k kS

Qualitative interpretation

V is switched on slowly (time scale ħ/ε) between -∞ and 0, and then switched off slowly (time scale ħ/ε) between 0 and +∞One starts from ϕi at t = -∞ and one looks for the probability amplitude to be in ϕj at t = +∞

0

0

From to the initial free state is transformed into Since the evolution operator is unitary, and since transforms into from to is the amplitude to find the system in the

ϕψ ψ

ϕ ψ ψ

+ −

− +

= −∞ =

= = +∞

,

.

,

i

i j

j j i

t t

t t free state at if

one starts from at ϕ

ϕ= +∞

= −∞.j

i

tt

(1.22)

(1.23)

16






17

Central potential

1D radial Schrödinger equationV depends only on r

One looks for solutions of the formϕ = =( ) ( ) ( ) /k l m k l l mr R r Y n n r r

If we put

0 0with the boundary condition

=

=

( )( )

( )

k lk l

k l

u rR r

ru

one gets for ukl the following 1D radial equation2

22 2 2

1 2 0dd

μ⎡ ⎤++ − − =⎢ ⎥

⎣ ⎦

( )()()k l

l lk V r u rr r

1D Schrödinger equation for a particle moving in a potential which is the sum of V and of the centrifugal barrier

2

2

12μ

+( )l lr

(1.24)

(1.25)

(1.26)

(1.27)

(1.28)

18

Case of a free particle (V=0)The solutions of the Schrödinger equation are:

20 2ϕ

π=()() ()()k l m l l m

kr j kr Y n

where the jl are the spherical Bessel functions of order l

0

12 1 2

π→ →∞

−+

()() () sin( )

( )!!

l

l lr r

krj kr j kr kr ll kr

For large r we thus have

0

2 2

2 2

22

outgoing spherical wave + ingoing spherical wave

π π

πϕπ

π

→∞

− − −

−

⎡ ⎤−⎣ ⎦=

=

()

( /) ( /)

sin( /)() ()

()

k l m l mr

i kr l i kr l

l m

kr lr Y nr

e eY n

ir

These functions form an orthonormal set (see Appendix)0 0ϕ ϕ δ δ δ′ ′ ′ ′ ′′= −()() ( )k l m klm ll mmk k

(1.29)

(1.30)

(1.31)

(1.32)

19

Expansion of a plane wave in free spherical waves

Plane wave3 22π δ− ′ ′= = −/.() ( )i k rr k e k k k k

The factor (2π) -3/2 is introduced for the orthonormalizationOne can show that:

3 2 3 2

0

0

0

2 2 4

1

with

π π π κ

κ ϕ

κ

∞ = +− −

= = −∞ =+

= = −

=

=

=

∑ ∑

∑ ∑

/. / *

* ()

() () ()()()()

()()()

/

m li k r l

lm lm ll m l

m ll

lm klml m l

e i Y n Y j kr

i Y rk

k k

{ }{ }

0

0

1

The transformation from the orthonormal basis to t

he orthonormal basis is given by the matrix

ϕ δ

ϕ

κ′ ′ ′ ′ ′′= −()

(

*

)

( ) ()k l m l m

klm

k k kk

k

Y

(1.33)

(1.34)

(1.35)

20

Effect of a potential. Phase shifts

We come back to the Schrödinger equation with V≠0.Consider, for r large, an incoming wave exp[-i(kr-lπ/2)]. Since the reflection coefficient of V is 1 (conservation of the norm), the reflected outgoing wave has the same modulus and has just accumulated a phase shift with respect to the V=0 case. The superposition of the 2 waves is thus a shifted sinusoid.

We conclude that there is a set of solutions of the Schrödinger equation with V≠0 which behave for large r as:

22 π δϕ

π→∞

⎡ ⎤− +⎣ ⎦sin / ()() ()

lklm lmr

kr l kr Y n

rOne can show that these functions are orthonormalized (see Appendix)

ϕ ϕ δ δ δ′ ′ ′ ′ ′′= −( )k l m klm ll mmk k (1.37)

They don’t form a basis if there are also bound states in the potential V.

(1.36)

21

Partial wave expansion of the outgoing scattering state

0

Consider the linear superposition of the states with the samecoefficients as those appearing in the expansion (1.34) of the plane wave on the , each state being multiplied by the phase fac

klm

klm

ϕ

ϕ()

3 22

tor We will show that such a state is nothing but the outgoing

state (multiplied by for having an orthonormalized state)l

k

iδψ π+ − /

exp().

()

0

0

0

1 e

e

δ

δ

ψ κ ϕ

ϕ ϕ

∞ = ++

= =−

∞ =+

= = −

=

=

∑ ∑

∑ ∑

*

()

()()l

l

m lil

lm klmkl m l

m li

klm klml m l

i Yk

k

0

Before demonstrating this identity, let us discuss its physicalmeaning.The outgoing scattering state is obtained by switching on slowly V on the free state. Each spherical wave of the expansion

()

klmϕ of is transformed into , but in addition it acquires

a phase factor e which depends on and which thus varies from one spherical wave of the expansion to another one.

l

klmi

klδ

ϕ

(1.38)

22

DemonstrationFor large r, the linear superposition introduced in (1.38) behaves as:

( ) ( )

( ) ( ) ( )

( )

2 2 2

02

2 2

0

2 2 2 1

1 22

11 2 22

1Using we g

e e

e e e

eet:

π

δ

δ

δ

π

π

κπ

δ

κ

π

ππ

π

− + − −∞ =

−∞ = +

= −

+

= =

=

−

⎡ ⎤−−⎢ ⎥+⎢

⎡ ⎤− + = − ×

⎣ ⎦

+⎣

−

− ⎦

⎥∑

∑

∑

∑

/

/

*

/

*

exp

()

sin( /)()() (

/ exp

(

)

()

/

)

l

l

l

il

i kr l i kr lm lll

i il ikrm lllm lm

m lml m l

l m l

i Y Y nk ir

kr li Y Y nk r i r

i kr l i kr l

The contribution of the first term of the bracket is nothing but the asymptotic expansion of the plane wave in spherical waves. The second term gives an outgoing spherical wave

3 22 eeπ κ− ⎡ ⎤+⎢ ⎥

⎣ ⎦/ .

( ) (,,)

ikri k r f k n

rThis demonstrates that the state given in (1.38) is an outgoing scattering state and gives in addition the expression of the amplitude f

(1.42)

(1.39)

(1.40)

(1.41)

23

Scattering amplitude in terms of the phase shifts

0

2

0

4

1 2 1

where is a Legendre polynomial and where is the angle between and Integrating over the polar an es

e

e

gl

δ

δ

θ θ

πκ δ κ

δ θ

κ

∞ = +

= =−∞

=

=

= +

∑ ∑

∑

*(,,) sin

(cos)

.

()()

( ) sin (cos)

l

l

m li

l lm lml m l

il l

l

l

f k n Y n Yk

l Pk

Pn f

of gives the scattering cross sectionκ2

20

4 2 1πσ σ σ∞

=

= = +∑() () () ( )sin()l ll

k k k lk

δ l k

Scattering of 2 identical particles

θ π - θ1 1 for bosons (fermions)

() () ( )

()k k kf f fθ θ ε π θ

ε→ + −

= + −22

02total d

πσ π θ θ θ ε π θ= +∫

/sin () ( )k kf f − (1.45)

Quantum interference between2 different paths

(1.43)

(1.44)

24

Partial wave expansion of the ingoing scattering state

is given by a linear superposition of the states analogous to the one introduced for each state being now multiplied by the phase factor instead of exp() exp().

klmk

k

l li i

ψ ϕψ

δ δ

−

+

−

0

0 0

1 e eδ δψ κ ϕ ϕ ϕ∞ = + ∞ = +

− −−

= =− = = −

= =∑ ∑ ∑ ∑* ()()()l l

m l m li il

lm klm klm klmkl m l l m l

i Y kk

0If we start from this state and if we switch off V slowly, it transforms into the free state . Each wave is transformed into , but in addition its phase factor changes from e to 1 w

()

l

klm klmi

kδ

ϕ ϕ− hich corresponds

to acquiring a phase factor e liδ+

0 0

2

Finally, when we go from to switching on and then switching off V slowly, we start from and we end in acquiring a global phase factor e e e

() ()

,

l l l

klm klmi i i

t t

δ δ δϕ ϕ

+ + +

The demonstration of this identity is similar to the one given above for the outgoing scattering state

= −∞ = +∞

× =

(1.46)

25

S – Matrix in the angular momentum representation

ψ ψ− +′ ′

′= =k k k kS k S k

0

0

e() l

m li

k l m k l mkl m l

k δψ ϕ ϕ′ ′∞ = +

−−′ ′ ′ ′ ′ ′′

′ ′ ′= = −

′= ∑ ∑0

0

e δψ ϕ ϕ∞ = +

+

= = −

= ∑ ∑ () l

m li

klm klmkl m l

k

0 0

0 0

e eδ δ

δ δ δ

ψ ψ ϕ ϕ ϕ ϕ

′ ′

′ ′∞ = + ∞ = ++− +

′ ′ ′ ′ ′ ′′ ′′ ′ ′= = − = = −

′−

′= = ∑ ∑ ∑ ∑ () ()

( )

l l

l l m m

m l m li i

k l m k l m klm klmk k k kl m l l m l

k k

S k k

0 0 0 0

0 0ϕ ϕ ϕ ϕ

′ ′∞ =+ ∞ = +

′ ′ ′ ′ ′ ′′′ ′ ′= = − = = −

′ ′= = ∑ ∑ ∑ ∑ () () () ()m l m l

k l m k l m klm klmk kl m l l m l

S k S k k S k

20 0 δϕ ϕ δ δ δ+′ ′ ′ ′ ′′= −e() ()

( )li

k l m klm l l m mS k k

We use the expansion of and in spherical wavesk k

ψ ψ+ −′

This gives a first expression of k kS ′

On the other hand, a change of basis gives for k kS ′

Comparing the 2 expressions obtained for we get,k kS ′

(1.48)

(1.49)

(1.50)

2which shows that the -matrix is diagonal in the angular momentumrepresentation, with diagonal elements clearly showingthe unitarity of

exp()l

Si

Sδ

(1.47)

(1.51)

26






27

Central potential. Low energy limit

Suppose first V=0. The centrifugal barrier in the 1D Schrödinger equation prevents the particle from approaching near the region r=0

2

2

12μ

+( )l lr

2 2 2μ/klr r

2 2 2

2

122

1

1 dB

μμ+

=

= +

= +

( )

( )

( )

l

l

l

l l kr

k r l l

r l l

If the range b of the potential is small enough, i.e. if

00

dB

a particle with cannot feel the potentialOnly wave will feel " -wave scattering"

≠

= .

b

ll V s

22

02 2

2 0dd

μ⎡ ⎤+ − =⎢ ⎥

⎣ ⎦()()kk V r u r

r

(1.52)

(1.53)

28

Scattering length0 0

0

For large enough, varies as Let be the function extending for all Let be the intersection point of with the -axis which is the closest from the origi

δδ

⎡ ⎤= +⎣ ⎦⎡ ⎤+⎣ ⎦

sin ()

sin () .k k

k k

k

r u u kr kv kr k u rP v r

0n.By definition, the scattering length a is the

limit of the abscissa of when (see figure)→P k

0 0 0

0

000

0

2 2

1

0Expansion of in powers of near

Abscissa of :

δ π δ

δ δ δ

δ

π

→

→

−= − ≤

⎡ ⎤= + →=

+⎣

−

+

⎦

≤

()sin () sin() cos()

tan()

tan()lim ()

k kr

k

k

v r kr k k kr

ka k

k

v kr krk

kk

P(1.54)

(1.55)

29

Scattering length (continued)Limit k=0

2

02 2

2 0dd

μ⎡ ⎤− =⎢ ⎥

⎣ ⎦()()V r u r

r

Far from r=o, the solution of the S.E. is a straight line and

0 ∝ −()v r r aThe abscissa of Q is equal to a

Scattering cross section 2

2 002 2

0 02

0

20

4 2 1

4

8

4

For identical bosons

δπ

σ

σ δ σ π

δ σ π

π

=

=

=→

=

−

=

+ ⇒ =

⇒ =

(

sin()() ( )sin() ()

() (

)

)

l l l

lk

l k

kk l k k

kk a k a

a

kk

(1.56)

(1.57)

(1.58)

(1.60)

(1.59)

30

Scattering length for square potentialsSquare potential barriers

()V r

b r

0 ∝ −()v r r a

a0

0V

0()u r

2 20 0 2

Square barrier ofheight and width

/V kb

μ=

0 0

0For and > == ∝ −( () )u vr r

r b kr a

0

20 0

0

0

0

0 0

For and

The curvature of ispositive and = =

==

′′<

(

(

)

) ()r b k

u

kur

u r u r

0

We conclude that the scattering length is always positive and smaller than the range of the potential

≤ ≤b

a b

0When (hard sphere potential) →

→ ∞Va b

31

Scattering length for square potentials (continued)Square potential wells

()V r

b r

0 ∝ −()v r r a

a 0

0V

0()u r

2 20 0 2μ= − /V k

0 0

0For and () ()urr

kav

brr

> == ∝ −

0

20 0

0

0

0

0 0

For and

The curvature of isnegative and

< =

= =

′′ = −() ()

( )

r b k

u

kur

u r u r

0

0

If is small enough so that there is no bound state in the potential well, the curvature of for is small and is negative<V

u r b a

0 0When increases, the curvature of for increases inabsolute value and Then switches suddenly to and decreases. This divergence of corresponds to the appearanceof the first bound state i

<→ −∞ + ∞.

V u r ba a

an the potential well

32

Square potential wells (continued)

Variations of a with k0figure taken from Ref.5

0 0 2 1 2When the depth of the potential well increases, divergencesof occur for all values of such that corresponding to the appearances of successive bound statesin the potential well.These

π= +( )/a V k b n

divergences of which goes from to arecalled "zero-energy" resonances

− ∞ + ∞a

33

Long range effective interactions and sign of aThe scattering length determines how the long range behavior of the wave functions is modified by the interactions. To understand how the sign of a is related to the sign of the effective long rangeinteractions, it will be useful to consider the particle enclosed in a spherical box with radius R, so that we have the boundary condition

0 0=()u Rleading to a discrete energy spectrumIn the absence of interactions (V=0), the normalized eigenstates and the eigenvalues of the 1D Schrödinger equation are:

2 2 20

2

1 1 22 2

π πψπ μ

= = =() sin( /)() ,,...N N

N r R Nr E NR r R

R r0

πsin( /)N r R(1.62)

Figure corresponding to N=3

(1.61)

34

R

0=V

Ra

00

≠>

Va

Ra

00

≠<

Va

The dotted line is the sinusoid outside the range of the potentialIt has a shorter wavelength than for V=0, and thus a larger wave number k.The kinetic energy in this region, which is also the total energy, is larger

The dotted line is the sinusoid outside the range of the potentialIt has a longer wavelength than for V=0, and thus a smaller wave number k.The kinetic energy in this region, which is also the total energy, is smaller

35

Correction to the energy to first order in a

Ra

2

0

2 2 2 2 2

2 22 2

2 1

2Because of the interactions, these half wavelengths occupynow a

For the state , we have

length so tha

tλ λ

π λ π

λ

λ

ψ

μ

−′= → = −′ ′= → = =

′ ′= → = = −

=

−

()

/ / /

/ ( )// / /(

/

)

)

(N N N N

N

N

NR

E k E E k k E R R

aR N

a

R a Nk k R

N

R a

E

R

k2⎛ ⎞

+⎜ ⎟⎝ ⎠

aR

2 2 2

3

2Finally, we have πδμ

′= − = =N N N N

a NE E E E aR R

Long range effective interactions are - repulsive if a > 0- attractive if a < 0

(1.63)

36






37

Model used for the potential V(r)Why not using the exact potential?The interaction potential is very difficult to calculate exactly.A small error in V can introduce a very large error on the scattering length deduced from this potential.Mean field description of ultracold quantum gases require in general a first order treatment of the effect of V (Born approximation). But Born approximation cannot be in general applied to the exact potential

Approach followed hereThe motivation here is not to calculate the scattering length. This parameter is supposed known experimentally. We are interested inthe derivation of the macroscopic properties of the gas from a mean field description using a single parameter which is a.The key idea is to replace the exact potential by a “pseudo-potential”simpler to use than the exact one and obeying 2 conditions:

- It has the same scattering length as the exact potential- It can be treated with Born approximation so that mean field

descriptions of its effects are possible

38

Determination of the pseudo-potential

We add to the 3D Schrödinger equation of a free particle (V=0) a term proportional to a delta function

2 2 2

2 2ψ δ ψ

μ μ− Δ + =() () ()

kr C r r

0 0

To determine the coefficient , we impose to the solution of this equation to coincide with the extension to all of the asymptotic behavior of the true wave function In parti

δ⎡ ⎤+⎣ ⎦sin ()/ ()/

Cr

kr k r u r r

0

cular, for small enough, one should have:ψ

→−() ( )/

rr

kr B a r

2

1 4

42

Inserting (1.65) into 1.64 and using we get an equation containing a delta function multiplied by a coefficient

which must vanish. This gives the coefficient appearing in (1.

π δ

πμ

Δ = −

−

( ) (/) (),r r

C a B

C 64)2 24 4

2where π π

μ= = =C g B g a a

m

(1.64)

(1.65)

(1.66)

Derivation “à la” H.Bethe and R.Peierls (Y.Castin, private communication)

39

Determination of the pseudo-potential (continued)It will be more convenient to express C=gB, not in terms of the coefficient B appearing in the wave function ψ = B(r-a)/r of equation (1.65), but in terms of the wave function ψ itself. We use for that

0

dd

ψ=

⎡ ⎤= ⎢ ⎥

⎣ ⎦rB r

rEquation (1) can be rewritten as:

[ ]

2 2 2

2 2ψ ψ ψ

μ μ

ψ δ ψ

− Δ + =

=

( ) ( ) ( )

( ) ( ) ( )

pseudo

pseudodwhere d

kr V r r

V r g r r rr

10 0

pseudo

pseudo

is called the pseudo-potential. The term d d regularizesthe action of when it acts on functions behaving as near

For functions which are regular in , has the same

δ⎡ ⎤⎣ ⎦

= =

(/)

() /.

V r rr r

r r V

0 0 00 0

pseudo

pseudo

effect as with

regular in

δψ ψ δψ ψ ψ δ

′= ≠ ⇒ == ⇒ =

():()()/ () () ()()

() () ()()

g rr u r r u V r g u rr r V r g r

(1.67)

(1.68)

(1.69)

(1.70)

40

Scattering states of the pseudo-potentialWe are looking for solutions of equation (1.68) with E > 0

00 0 0

0pseudo pseudo

For the centrifugal barrier prevents the particle fromapproaching and one can show that so that,according to (1.70), gives only s-scatteringand one can write ( )

ψψ

ψ

≠= =

=

,

, (),

().

lr

V r Vr 0 0 0 0where can be = ≠()/ () .u r r u

20 0 0 0 0

0 2

0 0 14 0dd

π δ⎡ ⎤−

Δ = Δ + = − +⎢ ⎥⎣ ⎦

() () () ()()()

u r u u r u uu r

r r r r rThe Schrödinger equation for u0 becomes:

2 2 20 0

0 04 0 02 2

Cancelling the term proportional to and the term independantof we get 2 equations:

π δ δμ μ

δδ

⎡ ⎤′′′⎢ ⎥− − + + =

⎢ ⎥⎣ ⎦

() ()()() ()()

()

(),

u r u rku r g r ur r

rr

200 02

0

00

20

μπ

′′= − = − + =′()

() ()()

ug a u r k u r

u(1.73)

(1.71)

(1.72)

41

Scattering states of the pseudo-potential (continued)

The solution of the second equation can be written:( )0 0δ= +()sinu r k r

Inserting this solution into the first equation gives:

0δ = −tan k aOn the other hand, the s-wave scattering amplitude is equal to:

00 0

1 e δ δ=() sinif k

kUsing equation (1.75) giving tanδ0 finally gives after simple algebra:

0 1= −

+()

af ki k a

0

pseudo pseudois proportional to A first order treatment of thus gives the correct result for the scattering amplitude in the zero energy limit ( This shows that Born approximation can be used

=

.

).

V a V

k apseudowith for ultracold atomsV

(1.77)

The 2 conditions imposed above on Vpseudo are thus fulfilled

(1.74)

(1.75)

(1.76)

42

The unitary limit

From the expression of the scattering amplitude obtained above, we deduce the scattering amplitude for identical bosons

2

2 2

81

πσ =+

()akk a

which is valid for all k.

2

1

1

8

The low energy limit gives the well known result: ( )σ π

( )

ka

ka

k a

21

18

There is another interesting limit, corresponding to high energy, or strong interaction leading to result independent of :

( ) πσ

( )

ka

ka a

kk

This is the so called “unitary limit”

(1.80)

(1.78)

(1.79)

43

Bound state of the pseudo-potential

2 2 2 22 2The calculation is the same as for the scattering states, except that we replace the positive energy by a negative one -The 2 equations derived from the Schrodinger equation are now:

/ /k μ κ μ

200 0

0

00

0κ′′= − − =

′()

() ()()

ua u r u r

uThe solution of the second equation (finite for r→∞) is:

0 e κ−=() ru rwhich inserted into the first equation gives:

1κ = /aThe pseudo-potential thus has a bound state with an energy

2

22μ= −E

aand a wave function:

⎛ ⎞−⎜ ⎟

⎝ ⎠exp

ra

(1.81)

(1.82)

(1.83)

(1.84)

(1.85)

44

Energy shifts produced by the pseudo-potential

We come back to the problem of a particle in a box of radius R.We have calculated above the energy shifts of the discrete energy levels of this particle produced by a potential characterized by a scattering length a. To first order in a, we found:

2 2 2

3 πδ

μ=N

NE aR

This result was deduced directly from the modification induced by the interaction on the asymptotic behavior of the wave functionsand not from a perturbative treatment of V. We show now that:

0 0pseudo pseudoto first order in δ ψ ψ= () ()

N N NE V V

which is another evidence for the fact that the effect of the pseudopotential can be calculated perturbatively, which is not the case for the real potential. For example, a hard core potential (V=∞ for r< a) cannot obviously be treated perturbatively, but its scattering length is a , and using a pseudo-potential with scattering length a allowsperturbative calculations.

(1.86)

(1.87)

45

Demonstration

0 12

πψπ

=() sin( /)()N

N r RrR r

The unperturbed normalized eigenfunctions of the particle in the spherical box are:

03 2

0

1 10

02

is regular in and

ψ ππ

ψ =

=

(

/

)

()

()

()

N

N

r

NR

r

0 0 0ψ δ ψ=so that

() ()() ()()pseudo N NV r g r

2 2 2 2 2203 30

2π πδ ψ

π μ= = =

We deduce that ()()N N

N NE g g aR R

which coincides with the result obtained above in (1.63).

(1.47)

(1.88)

(1.89)

(1.90)

46

Conclusion

Elastic collisions between ultracold atoms are entirely characterized by a single number, the scattering lengthEffective long distance interactions are attractive if a<0 and repulsive if a>0Giving the same scattering length as the real potential, the pseudo-potential gives the good asymptotic behavior for the wave function describing the relative motion of 2 atoms, and thus correctly describes their long distance interactionsIn a dilute gas, atoms are far apart. The pseudo-potential is proportional to a and can be treated perturbatively. A first order treatment of the pseudo-potential is the basis of mean field description of Bose Einstein condensates where each atom moves in the mean field produced by all other atoms.Next step: can one change the scattering length?

atom-atom interactions in ultracold quantum gasescastin/cohen1.pdf · 1 collège de france...

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